Question : If the sides of an equilateral triangle are increased by 1 metre, then its area is increased by $\sqrt3$ sq. metre. The length of any of its sides is:

Option 1: $2$ metres

Option 2: $\frac{5}{2}$ metres

Option 3: $\frac{3}{2}$ metres

Option 4: $\sqrt3$ metres

Question : The radius of the base and curved surface area of a right cylinder are $r$ units and $4\pi rh$ square units respectively. The height of the cylinder is:

Option 1: $\frac{h}{2}$ units

Option 2: $h$ units

Option 3: $2h$ units

Option 4:  $4h$ units

Question : If $x>1$ and $x+\frac{1}{x}=2\frac{1}{12}$, then the value of $x^{4}-\frac{1}{x^{4}}$ is:

Option 1: $\frac{58975}{20736}$

Option 2: $\frac{59825}{20736}$

Option 3: $\frac{57985}{20736}$

Option 4: $\frac{57895}{20736}$

Question : If $x\cos \theta -y\sin \theta =\sqrt{x^{2}+y^{2}}$ and $\frac{\cos ^2{\theta }}{a^{2}}+\frac{\sin ^{2}\theta}{b^{2}}=\frac{1}{x^{2}+y^{2}},$ then the correct relation is:

Option 1: $\frac{x^{2}}{b^{2}}-\frac{y^{2}}{a^{2}}=1$

Option 2: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

Option 3: $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$

Option 4: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

Question : If $x^4+\frac{1}{x^4}=194, x>0$, then find the value of $x^3+\frac{1}{x^3}+x+\frac{1}{x}$

Option 1: 76

Option 2: 66

Option 3: 56

Option 4: 46